# Operations on Root Data ## `R1 eq R2: RootDtm, RootDtm -> BoolElt` Returns `true` if, and only if, $R_1$ and $R_2$ are identical root data. ## `IsIsomorphic(R1, R2): RootDtm, RootDtm -> BoolElt, [RngIntElt], Map` Returns `true` if, and only if, $R_1$ and $R_2$ are isomorphic root data. If `true`, the second value returned is a sequence giving the simple root of $R_2$ corresponding to each simple root of $R_1$, and the third value returned is an isomorphism $R_1\to R_2$. This function is currently only implemented for semisimple root data. ## `IsCartanEquivalent(R1, R2): RootDtm, RootDtm -> BoolElt, SeqEnum` Returns `true` if and only if the root data $R_1$ and $R_2$ are Cartan equivalent; i.e. they have isomorphic Dynkin diagrams. If `true`, the second value returned is a sequence giving the simple root of $R_2$ corresponding to each simple root of $R_1$. ## `IsIsogenous(R1, R2): RootDtm, RootDtm -> BoolElt, SeqEnum, RootDtm, Map, Map, RootDtm, Map, Map` Returns `true` if, and only if, $R_1$ and $R_2$ are isogenous root data. If `true`, the subsequent values returned are: a sequence giving the root of $R_2$ corresponding to each root of $R_1$, the corresponding adjoint root datum $R_{ad}$, the morphisms $R_{ad}\to R_1$ and $R_{ad}\to R_2$, the corresponding simply connected root datum $R_{sc}$, and the morphisms $R_1\to R_{sc}$ and $R_2\to R_{sc}$. ## `Example: Isomorphism Isogeny (ex-864e8f)` An example of isogenous root data: ```magma > R1 := RootDatum("A3"); > R2 := RootDatum("A3" : Isogeny := "SC"); > R1 eq R2; false > IsIsomorphic(R1, R2); false > IsCartanEquivalent(R1, R2); true [ 1, 2, 3 ] > IsIsogenous(R1, R2); true [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 ] Adjoint root datum of type A3 Mapping from: RootDtm: ad to RootDtm: ad Mapping from: RootDtm: ad to RootDtm: sc Simply connected root datum of type A3 Mapping from: RootDtm: ad to RootDtm: sc Mapping from: RootDtm: sc to RootDtm: sc ``` An example of distinct isomorphic root data: ```magma > C := CartanMatrix("B2"); > R1 := RootDatum(C); > R2 := RootDatum(Transpose(C)); > R1; R2; Adjoint root datum of type B2 Adjoint root datum of type C2 > R1 eq R2; false > IsIsomorphic(R1, R2); true [ 2, 1 ] ``` ## `CartanName(R): RootStr -> MonStgElt` The Cartan name of the root datum $R$ (Section [Finite and Affine Coxeter Groups](../CoxeterSystems/finiteaffine.md#sectcartanfinaff)). ## `TwistedCartanName(R): RootDtm -> MonStgElt` The twisted Cartan name of the root datum $R$. E.g., `"2A3,2"`. ## `CoxeterDiagram(R): RootStr` Print the Coxeter diagram of the root datum $R$ (Section [Finite and Affine Coxeter Groups](../CoxeterSystems/finiteaffine.md#sectcartanfinaff)). ## `DynkinDiagram(R): RootStr` Print the Dynkin diagram of the root datum $R$ (Section [Finite and Affine Coxeter Groups](../CoxeterSystems/finiteaffine.md#sectcartanfinaff)). ## `CoxeterMatrix(R): RootStr -> AlgMatElt` The Coxeter matrix of the root datum $R$ (Section [Coxeter Matrices](../CoxeterSystems/coxetermat.md#sectcartancoxmat)). ## `CoxeterGraph(R): RootStr -> GrphUnd` The Coxeter graph of the root datum $R$ (Section [Coxeter Graphs](../CoxeterSystems/coxetergraph.md#sectcartancoxgrph)). ## `CartanMatrix(R): RootStr -> AlgMatElt` The Cartan matrix of the root datum $R$ (Section [Cartan Matrices](../CoxeterSystems/cartanmat.md#sectcartancarmat)). ## `DynkinDigraph(R): RootStr -> GrphDir` The Dynkin digraph of the root datum $R$ (Section [Dynkin Digraphs](../CoxeterSystems/dynkindigraph.md#sectcartandyndigrph)). ## `Example: Diagrams (ex-de9285)` ```magma > R := RootDatum("F4"); > DynkinDiagram(R); F4 1 - 2 =>= 3 - 4 > CoxeterDiagram(R); F4 1 - 2 === 3 - 4 ``` ## `GammaAction(R): RootDtm -> Rec` The $\Gamma$-action of the root datum $R$. This is a record consisting of four elements: `gamma` is the Group $\Gamma$ acting on $R$, `perm_ac` is the homomorphism defining the permutation action of $\Gamma$ on the set of all roots of $R$, finally `mats_rt` and `mats_co` are sequences of matrices defining the action of $\Gamma$ on the root and coroot spaces of $R$. ## `GammaRootSpace(R): RootDtm -> GSetEnum, Map` ## `GammaCorootSpace(R): RootDtm -> GSetEnum, Map` Given a root datum $R$, create the fixed space of $\Gamma$ acting on the (co)root space $V = {\mathbb{Q}}\otimes X$ of $R$ as well as the embedding in $V$. ## `GammaOrbitOnRoots(R, r): RootDtm, RngIntElt -> GSetEnum` The orbit through the $r$th root of the $\Gamma$-action on the root datum $R$. ## `GammaOrbitsOnRoots(R): RootDtm -> SeqEnum[GSetEnum]` ## `PositiveGammaOrbitsOnRoots(R): RootDtm -> SeqEnum[GSetEnum]` ## `NegativeGammaOrbitsOnRoots(R): RootDtm -> SeqEnum[GSetEnum]` ## `ZeroGammaOrbitsOnRoots(R): RootDtm -> SeqEnum[GSetEnum]` The sequence of all (respectively positive, negative and zero) orbits of the $\Gamma$-action on the root datum $R$ (Section [Extended Root Data](introduction.md#subsectextrd)). ## `GammaActionOnSimples(R): RootDtm -> HomGrp` The $[\Gamma]$-action on the simple (co)roots of the root datum $R$. (Section [Extended Root Data](introduction.md#subsectextrd)). This function was called `GammaActionPi` in the last release. ## `OrbitsOnSimples(R): RootDtm -> SeqEnum[GSetEnum]` The sequence of all orbits of the $[\Gamma]$-action on the simple (co)roots of the root datum $R$ (Section [Extended Root Data](introduction.md#subsectextrd)). This function was called `OrbitsPi` in the last release. ## `DistinguishedOrbitsOnSimples(R): RootDtm -> SeqEnum[GSetEnum]` The sequence of distinguished orbits of the $[\Gamma]$-action on the simple (co)roots of the root datum $R$ (Section [Extended Root Data](introduction.md#subsectextrd)). This function was called `DistinguishedOrbitsPi` in the last release. ## `BaseRing(R): RootDtm -> RngInt` The base ring of the root datum $R$ is the field of rational numbers. ## `Rank(R): RootStr -> RngIntElt` ## `AbsoluteRank(R): RootDtm -> RngIntElt` The (absolute) rank of the root datum $R$, i.e. the number of simple (co)roots. ## `RelativeRank(R): RootDtm -> RngIntElt` The relative rank of the root datum $R$, i.e. the number of simple (co)roots of the relative root system. This is the same as absolute rank for split root data. ## `Dimension(R): RootStr -> RngIntElt` The dimension of the root datum $R$, i.e. the dimension of the (co)root space. This is at least as large as the rank, with equality when $R$ is semisimple. ## `TwistingDegree(R): RootDtm -> RngIntElt` The twisting degree of the root datum $R$, i.e. the order of $\Gamma$ divided by the kernel of the $[\Gamma]$-action. ## `AnisotropicSubdatum(R): RootDtm -> RootDtm` The anisitropic subdatum of the root datum $R$. ## `Example: Operations For Twisted Root Data (ex-9c2a05)` Consider the twisted root datum of type ${}^2A_{3,1}$ with distinguished orbit $\{2\}$: ```magma > R := RootDatum( "A3" : Twist := < {{2}}, 2 > ); ``` First, print out the action of $\Gamma$ on the root datum: ```magma > GammaAction(R); rec | gamma := Permutation group acting on a set of cardinality 4 Order = 4 = 2^2 (1, 2, 3, 4), perm_ac := Homomorphism of GrpPerm: $, Degree 4, Order 2^2 into GrpPerm: $, Degree 12, Order 2^10 * 3^5 * 5^2 * 7 * 11 induced by (1, 2, 3, 4) |--> (1, 3, 7, 9)(2, 4, 6, 5)(8, 10, 12, 11), mats_rt := [ [ 0 0 1] [ 1 1 0] [-1 0 0] ], mats_co := [ [ 0 0 1] [ 0 1 0] [-1 1 0] ] > ``` Compute the orbits of the $\Gamma$-action: ```magma > PositiveGammaOrbitsOnRoots(R); [ GSet{ 2, 4, 5, 6 } ] > NegativeGammaOrbitsOnRoots(R); [ GSet{ 8, 10, 11, 12 } ] > ZeroGammaOrbitsOnRoots(R); [ GSet{ 1, 3, 7, 9 } ] > &+[ Root(R,r) : r in ZeroGammaOrbitsOnRoots(R)[1] ]; (0 0 0) ``` Compute the $[\Gamma]$-action and its orbits: ```magma > GammaActionOnSimples(R); Homomorphism of GrpPerm: $, Degree 4, Order 2^2 into GrpPerm: $, Degree 3, Order 2 * 3 induced by (1, 2, 3, 4) |--> (1, 3) > OrbitsOnSimples(R); [ GSet{ 2 }, GSet{ 1, 3 } ] > DistinguishedOrbitsOnSimples(R); [ GSet{ 2 } ] ``` Absolute and relative rank and the twisting degree, as well as their appearance in the name of the root datum: ```magma > AbsoluteRank(R); 3 > RelativeRank(R); 1 > TwistingDegree(R); 2 > R; R: Twisted adjoint root datum of type 2A3,1 ``` anisotropic subdatum: ```magma > A := AnisotropicSubdatum(R); A; A: Twisted root datum of type 2(A1 A1)2,0 > GammaAction(A)`perm_ac; Homomorphism of GrpPerm: $, Degree 4, Order 2^2 into GrpPerm: $, Degree 4, Order 2^2 induced by (1, 2, 3, 4) |--> (1, 2, 3, 4) ``` ## `CoxeterGroupOrder(R): RootStr -> RngIntElt` The order of the (split) Coxeter group of the root datum $R$. ## `GroupOfLieTypeOrder(R, q): RootDtm, RngElt -> RngIntElt` The order of the group of Lie type with split root datum $R$ over the field of cardinality $q$. ## `GroupOfLieTypeFactoredOrder(R, q): RootDtm, RngElt -> RngIntElt` The factored order of the group of Lie type with split root datum $R$ over the field of order $q$. ## `Example: Group Of Lie Type Order (ex-da6059)` As well as accepting a specific prime power, these functions also take an indeterminate so that the generic order formula can be computed. ```magma > P := PolynomialRing(Integers()); > R := RootDatum("F4"); > GroupOfLieTypeFactoredOrder(R, q); [ , , , , , , , ] > > R := RootDatum("B2"); > ord := GroupOfLieTypeOrder(R, q); > forall{ q : q in [2..200] | not IsPrimePower(q) or > Evaluate(ord, q) eq GroupOfLieTypeOrder(R, q) }; true ``` ## `FundamentalGroup(R): RootDtm -> GrpAb, Map` The fundamental group $\Lambda/{\mathbb{Z}}\Phi$ of the root datum $R$ together with the projection $\Lambda\rightarrow\Lambda/{\mathbb{Z}}\Phi$. See Subsection [Isogeny of Split Reduced Root Data](introduction.md#subsectrdisogeny). ## `IsogenyGroup(R): RootDtm -> GrpAb, Map` The isogeny group $X/{\mathbb{Z}}\Phi$ of the root datum $R$ together with the projection $X\rightarrow X/{\mathbb{Z}}\Phi$. If $R$ is semisimple, the injection $X/{\mathbb{Z}}\Phi\rightarrow\Lambda/{\mathbb{Z}}\Phi$ is also returned. See Subsection [Isogeny of Split Reduced Root Data](introduction.md#subsectrdisogeny). ## `CoisogenyGroup(R): RootDtm -> GrpAb, Map` The coisogeny group $Y/{\mathbb{Z}}\Phi^\star$ of the root datum $R$ together with the projection $Y\rightarrow Y/{\mathbb{Z}}\Phi^\star$. If $R$ is semisimple, the projection $Y/{\mathbb{Z}}\Phi^\star\rightarrow\Lambda/{\mathbb{Z}}\Phi$ is also returned. See Subsection [Isogeny of Split Reduced Root Data](introduction.md#subsectrdisogeny). ## `Example: Isogeny Groups (ex-2f8db1)` In the semisimple case, the fundamental group contains the isogeny group, with quotient isomorphic to the coisogeny group. ```magma > R := RootDatum("A5" : Isogeny := 3); > F := FundamentalGroup(R); > G := IsogenyGroup(R); > H := CoisogenyGroup(R); > #G * #H eq #F; true ``` Nonsemisimple root data have infinite isogeny groups. ```magma > R := StandardRootDatum("A", 5); > IsogenyGroup(R); Abelian Group isomorphic to Z Defined on 1 generator (free) ```